Interface crack between two shear deformable elastic layers

Abstract

An improved method based on the first-order shear deformable plate theory is developed to calculate the energy release rate and stress intensity factor for a crack at the interface of a bi-layer structure. By modeling the uncracked region of the structure as two separate Reissner–Mindlin plates bonded perfectly along the interface, this method is able not only to take into account the shear deformation in the cracked region, but also to capture the shear deformation in the uncracked region of the structure. A closed form solution of energy release rate and mode decomposition at the interface crack is obtained for a general loading condition, and it indicates that the energy release rate and stress intensity factor are determined by two independent loading parameters. Compared to the approach based on the classical plate theory, the proposed method provides a more accurate prediction of energy release rate as well as mode decomposition. The computational procedures introduced are relatively straightforward, and the closed form solution can be used to predict crack growth along the layered structures.

Introduction

Interface cracking is one of common failure modes in multi-layered structures. Typical examples include delamination of composite laminates, debonding of adhesive joints, and decohesion of thin films from substrates. Fracture mechanics principles have been widely employed to assess this type of failure mode, where the energy release rate (G) or stress intensity factor (K) is evaluated and compared with the critical mode-mix-dependent G_c or K_c of the interface determined from experiments. The interface crack is likely to propagate if G or K reaches G_c or K_c. This approach was adopted by Wang and Crossman (1980); Wang (1982); O'Brien 1982, O'Brien 1990; and Davidson et al. (2000). By noting the mode mix dependence of G_c and K_c, it is necessary to extract the mode mix of G and K at the crack tip in order to successfully predict the growth of crack.

Finite elements have been frequently used to calculate G or K and mode mix of the interface crack (Matos et al., 1989; Venkatesha et al., 1996; Beuth, 1996; Sun and Qian, 1997; Qian and Sun, 1998) for general conditions. When the beam/plate-type layered structures are encountered, however, this method is not efficient since the K-dominance zone is very small; therefore, very fine mesh is required to obtain sufficiently accurate results. In such a case, a more efficient alternative is to use a semi-analytical method, where the energy release rate is determined by analytical means in an asymptotic approximation and the mode mix is obtained by solving a 2-D continuum problem. This method is remarkably simple, and therefore, computationally efficient, as proposed and illustrated by Schapery and Davidson (1990) and Suo and Hutchinson (1990). In the original works of Schapery and Davidson (1990); Suo and Hutchinson (1990); Davidson et al. (1995), the energy release rate of an interface crack between two elastic layers was calculated by a classical beam or plate theory; but the mode mix was retrieved through either the finite element method (Schapery and Davidson, 1990; Davidson et al., 1995) or the integral equation method (Suo and Hutchinson, 1990). These methods, commonly known as crack tip element (CTE) analysis (Davidson et al., 1995), were successfully used in the interface fracture analysis (Davidson et al., 2000). However, the shear deformation in the cracked and uncracked regions is not considered in the existing model since the classical beam or plate theory was basically used. As a result, the energy release rate is always underestimated by this method. Comparing with the finite element analysis results, Davidson and Sundararaman (1996) found that the maximum error in the CTE prediction of energy release rate could be as large as 13.7% for an isotropic bi-material glass/epoxy specimen with a slenderness ratio of 8.33. The shear deformation effect on the energy release rate for anisotropic materials with relatively small transverse shear modulus such as polymer composite laminates is even more pronounced as evidenced by Bruno and Greco (2001), where the shear deformation was found to be more than half of the total energy release rate for an orthotropic double cantilever beam specimen. Therefore, it is necessary to account for the shear deformation in computation and prediction of the energy release rate, especially when the materials with relatively low transverse shear modulus and moderate thickness are concerned. A notable effort to incorporate the shear deformation into the energy release rate was made by Bruno and Greco (2001). They modeled the undelaminated region of the laminate as two Reissner–Mindlin plates bonded by a linear elastic interface, instead of only a single plate element as in Schapery and Davidson (1990), and an interface model was introduced to ensure the displacement continuity along the interface of the two plates. The energy release rate was recovered by taking the limit of strain energy per unit interface area at the crack tip, when the interfacial stiffness approaches infinite. Closed form solutions of the energy release rate for certain simple configurations were obtained. The coupling terms in their solutions revealed the obvious shear deformation effect on the energy release rate. However, the energy release rate for general configurations was obtained through a process of taking limits with the aid of numerical calculation, and only a global mode decomposition based on the classical plate theory was introduced in their paper.

The present study is a further effort to enhance the accuracy of the energy release rate and stress intensity factor evaluations of interface cracks between two shear deformable elastic layers, by taking the shear deformation into consideration through modeling the layered systems as Reissner–Mindlin plates. This approach can be traced back to the studies of Armanios (1984); Armanios et al. (1986); Chatterjee et al. (1986); and Chatterjee and Ramnath (1988) for sublaminate analysis. Noting that in Armanios's formulation, the concentrated crack tip forces were neglected, and therefore, extra considerations of continuity or boundary conditions at the crack tip are required.

The aim of this study is to establish a simple closed-form solution for the energy release rate and stress intensity factor that can capture the effect of shear deformation in the cracked and uncracked regions of layered structures. The solution should be essentially the same as the classical solution by Suo and Hutchinson (1990) or Schapery and Davidson (1990) when it is reduced to the simple case in which the shear deformation is neglected. In this sense, this study can be regarded as an extension of the work in the classical plate model to the one in the first-order shear deformation model.

This paper is organized as follows: the analytical work of a bi-layer plate system using the first-order shear deformation theory is first established, followed by the interface fracture analysis of which the explicit solution for the energy release rate and mode decomposition of a crack along the bi-layer interface is obtained. To validate the proposed work, the present solution is compared with the finite element analysis in which the shear deformation is accounted for.